Institute for Computational and Mathematical Engineering and ‡Department of Chemistry, Stanford University , Stanford, California 94305, United States.

Abstract

This article surveys the recent application of optical Fourier processing to the long-established but still expanding field of single-molecule imaging and microscopy. A variety of single-molecule studies can benefit from the additional image information that can be obtained by modulating the Fourier, or pupil, plane of a widefield microscope. After briefly reviewing several current applications, we present a comprehensive and computationally efficient theoretical model for simulating single-molecule fluorescence as it propagates through an imaging system. Furthermore, we describe how phase/amplitude-modulating optics inserted in the imaging pathway may be modeled, especially at the Fourier plane. Finally, we discuss selected recent applications of Fourier processing methods to measure the orientation, depth, and rotational mobility of single fluorescent molecules.

Schematic diagram of a 4f optical processing system. In a conventional microscope, light is collected by an objective lens, collimated, and relayed through the back focal plane of the objective. (The z direction is always assumed to lie parallel to the optical axis.) A tube lens focuses the collected light into an image at the “intermediate” image plane. The 4f system (red box) is inserted a distance f4f behind the intermediate image plane. The Fourier transformed electric field, Ebfp(x′,y′) (a scaled image of the back focal plane), is projected onto a phase mask, and the phase modulated field is then focused into an image on a detector by a second lens. The precise scaling of the back focal plane image incident upon the phase mask will depend upon the numerical aperture of the objective, and the magnification of the objective/tube lens pair (calculated from fobj and ftube).

Schematic of coordinate systems used to calculate the images formed from single molecule fluorescence. (a) Two equivalent parametrizations for expressing the orientation of a molecule’s transition dipole moment. Either a unit vector μ̂ or a pair of angles {Φ, Θ} is used. (b) Ray emanating from a molecule with trajectory defined by the unit vector r̂ having intensity Iff(ϕ,θ) ∝ sin2 (η), where η is the angle between μ̂ and r̂. The distribution of Iff(ϕ,θ) is thus a torus (pictured). (c) Approximations used for modeling defocus: If d ≪ r, then r′ ≈ r and θ′ ≈ θ. (d) Overview of the complete imaging system modeled by our simulations. Note that we assume θtube is small, and therefore the electric fields emerging from the tube lens will have a z-component that is nearly zero.

Simulation of single-molecule images and basis functions. (a) Overview of simulation: A molecule (λ = 600 nm) is translated a varying distance d from the objective’s focal plane in isotropic media. We specify that the objective has an immersion medium of n1 = 1.518, and an NA of 1.4 (θmax = 67.26°). (b) Simulated images for a molecule with dipole moment oriented at: {Φ, Θ} = {45°, 45°}. (c) Basis functions used to simulate the defocused image d = 1000 nm. Note that the intensity color scale varies for each basis function. Units of length are specified in object space, i.e., before accounting for the magnification imparted by the objective/tube lens combination.

Dual-polarization/4f optical processing system. Adapted from ref () with permission. (a) Schematic diagram of experimental setup. EP and ES denote P- and S-polarized electric fields with respect to the beamsplitter, which subsequently are separated into ET and ER, the fields present in the transmitted and reflected polarization channels, respectively. (b) Plot of the phase function defining the quadrated pupil. Axis along which incident light is polarized is also sketched. (c) Geometry of our setup ensures that both the R and T channels are polarized along a single axis, so that the SLM can properly modulate all light emitted by the specimen.

Representative data set adapted from ref () with permission. (a) Widefield image of single dye molecules. Both T and R channels are shown. Note that due to the geometry of the experimental setup, the R channel image undergoes an additional reflection before being projected onto the EMCCD. Inset: the pair of angles {Θ, Φ} denote a single point on the unit hemisphere. (b) Partitioning scheme used for processing measured and simulated data into the vectors γ̂meas and γ̂sim.

Invariance to defocus. (a) Representative simulated images of a single molecule {Θ = 40°, Φ = 25°}. (b) Plot of normalized entries of γ̂sim as a function of defocus d. In the ±150 nm range indicated, the components of γ̂sim change minimally. For this simulation, an isotropic medium was assumed (no index mismatch).

Measurement results adapted from ref () with permission. (a) and (b) Orientation measurements for two molecules. At left: raw data and simulated images obtained from the mean orientation estimate. Center: repeated orientation measurements for the same molecule, plotted on the unit hemisphere. The 2σ ellipse computed from the data-covariance matrix is plotted in green. Right: Magnified view of the region of interest. For the molecule in (a), the mean orientation was: {Θavg = 42.2, Φavg = 242.2} with a standard deviation of {σΘ = 1.8°, σΦ = 1.7°}. An average of 2370 photons were detected per exposure. For the molecule in (b), we found {Θavg = 73.9°, Φavg = 326.9°} and {σΘ = 5.8°, σΦ = 4.3°}. An average of 921 photons were detected. (c) Orientation measurements for a single molecule over a ±150 nm range. Standard deviations at each depth are depicted by blue bars. (d) Sample images taken at different focal planes demonstrate robustness to defocus. For this molecule, an average of 916 photons per exposure were detected.

Overview of the bisected pupil. Adapted from ref () with permission. (a) The bisected pupil phase mask is plotted, and the polarization axis of incident light is indicated. (b) Simulations of an isotropic emitter imaged with the bisected pupil, at varying depths (color scale has been renormalized for each d, to display fine features of the PSF). (c) Calibration curve relating lobe separation distance to depth. This calibration was acquired by translating the objective lens relative to a fluorescent bead. A polynomial curve fitted to the data is also shown, indicating a nearly linear relationship.

Rotational mobility measurements with a bisected pupil. Adapted from ref () with permission. (a) The “rotation within a cone” model: A molecule is assumed to have a mean orientation described by the pair of angles {Θ, Φ}, and may rotate to any orientation within the cone specified by the angle α. (b) Diagram indicating the regions of an image that are summed when the linear dichroism (LD) and lobe asymmetry (LA) of a molecule are calculated. (c) Histograms of simulated single molecule images (blue) indicate that rotational mobility is high. Experimentally acquired data (red) most closely matches simulation using α = 75°. Note that as rotational mobility increases, standard deviations σLA and σLD, of histogrammed data decrease. (d) Super-resolution images color coded according to LD and T-channel LA.

Schematic diagram of a molecule at an interface. Δ is the distance of the molecule from the interface. Δ > 0 indicates that the molecule is above the interface (in medium 2), and Δ < 0 indicates that the molecule is below the interface (in medium 1). d is the distance of the interface to the objective’s focal plane (d > 0 means that the interface is above the focal plane). The angles θ1 and θ2 indicate the polar trajectory of a given ray propagating from the molecule and are related by Snell’s law.